Wavelet approach to operator-valued Hardy spaces

Abstract

This paper is devoted to the study of operator-valued Hardy spaces via the wavelet method. This approach is parallel to that in the noncommutative martingale case. We show that our Hardy spaces defined by wavelets coincide with those introduced by Tao Mei via the usual Lusin and Littlewood–Paley square functions. As a consequence, we give an explicit complete unconditional basis of the Hardy space $H_1(\mathbb{R})$ when $H_1(\mathbb{R})$ is equipped with an appropriate operator space structure.